Avi Wigderson , Hebrew University at Jerusalem and Institute for Advanced Study

ABSTRACT

The main concrete result of this paper is the first explicit construction of constant degree "lossless" expanders. In these graphs, the expansion factor is almost as large as possible: (1-eps)D, where D is the degree and eps is an arbitrarily small constant. Such graphs are known to have many applications, e.g. in constructing networks that can implement fast distributed, routing algorithms, expander-based linear codes, various storage schemes, and hard tautologies for various proof systems. The best previous explicit constructions gave expansion factor D/2, which is too weak for many applications. The D/2 bound was obtained via the eigenvalue method, and is known that that method cannot give better bounds.The main abstract contribution of this paper is the introduction and initial study of "randomness conductors," a notion which generalizes extractors, expanders, condensers and other similar objects. In all these functions, certain guarantee on the input "entropy" is converted to a guarantee on the output "entropy". For historical reasons, specific objects used specific guarantees of different flavors (e.g., in expanders entropy means "support size", and their property is satisfied whenever input entropy is small. In contrast, in extractors, entropy means "min-entropy" and their property is satisfied whenever input entropy is large). We show that the flexibility afforded by the conductor definition leads to interesting combinations of these objects, and to better constructions such as those above.The main technical tool in these constructions is a natural generalization to conductors of the zig-zag graph product, previously defined for expanders and extractors.